The Nuclear Equation of State and the neutron skin thickness in nuclei

Transcription

1 The Nuclear Equation of State and the neutron skin thickness in nuclei Xavier Roca-Maza Università degli Studi di Milano e INFN, sezione di Milano Physics beyond the standard model and precision nucleon structure measurements with parity-violating electron scattering Trento, August 1st-5th

2 Table of contents: Brief introduction The Nuclear Many-Body Problem Nuclear Energy Density Functionals Nuclear Equation of State: the symmetry energy The neutron skin thinckess and the symmetry energy The impact of the neutron skin (symmetry energy) on nuclear and astrophyiscs observables Some examples: Neutron stars outer crust, parity violating asymmetry, pygmy states (?), GDR, dipole polarizability, GQR, AGDR... Conclusions 2

3 INTRODUCTION 3

4 The Nuclear Many-Body Problem: Nucleus: from few to more than 200 strongly interacting and self-bound fermions. Underlying interaction is not perturbative at the (low)energies of interest for the study of masses, radii, deformation, giant resonances,... Complex systems: spin, isospin, pairing, deformation,... Many-body calculations based on NN scattering data in the vacuum are not conclusive yet: different predictions (interaction in the medium) are found depending on the approach EoS and (recently) few groups in the world are able to perform extensive calculations for light and medium mass nuclei Based on effective interactions, Nuclear Energy Density Functionals are successful in the description of masses, nuclear sizes, deformations, Giant Resonances,... 4

5 Nuclear energy density functionals E[ρ] are commonly derived from an effective Hamiltonian solved at first order perturbation theory (Hartree-Fock) E = Ψ H Ψ Φ H eff (ρ) Φ Exact EDF? E[ρ] Kohn-Sham iterative scheme (static) Determine/copy/invent/derive... E[ρ] Initial guess ρ 0 Calculate potential V eff from ρ 0 (h = δe/δρ) Solve single particle equation of motion (hφ i = ǫ i φ i ) φ i A Use φ i for calculating new ρ 1 = φ i 2 Repeat until consistency between ρ and V eff Runge-Gross Theorem (dynamic): exist also E[ρ(t),t] dt{ Φ(t) i t Φ(t) E[ρ(t),t]} = 0 i Useful for the study of small perturbations of the gs ρ: GR 5

6 Nuclear Energy Density Functionals: Main types of successful EDFs for the description of masses, deformations, nuclear distributions, Giant Resonances,... Relativistic mean-field models, based on Lagrangians where effective mesons carry the interaction: L int = ΨΓ σ ΨΦ σ + ΨΓ δ τψφ δ ΨΓ ω γ µ ΨA (ω)µ ΨΓ ρ γ µ τψa (ρ)µ e Ψ ˆQγ µ ΨA (γ)µ Non-relativistic mean-field models, based on Hamiltonians where effective interactions are proposed and tested: VNucl eff = Vlong range attractive +V short range repulsive +V SO +V pair Fitted parameters contain (important) correlations beyond the mean-field Nuclear energy functionals are phenomenological not directly connected to any NN (or NNN) interaction 6

7 The Nuclear Equation of State: Infinite System neutron matter e(ρ,δ=1) e ( MeV ) 10 0 Saturation (0.16 fm 3, 16.0 MeV) S(ρ)~ -10 e(ρ,δ=0) symmetric matter ρ ( fm 3 ) E A (ρ,β) = E A (ρ,β = 0) + S(ρ)β2 +O(β 4 ) Nuclear Matter [ β = ρ ] n ρ p ρ 7

8 The Nuclear Equation of State: Infinite System neutron matter e(ρ,δ=1) e ( MeV ) 10 0 Saturation (0.16 fm 3, 16.0 MeV) S(ρ)~ -10 e(ρ,δ=0) symmetric matter ρ ( fm 3 ) E A (ρ,β) = E A (ρ,β = 0) + S(ρ)β2 +O(β 4 ) Nuclear Matter Symmetric Matter [ β = ρ ] n ρ p ρ 8

9 The Nuclear Equation of State: Infinite System neutron matter e(ρ,δ=1) e ( MeV ) 10 0 Saturation (0.16 fm 3, 16.0 MeV) S(ρ)~ -10 e(ρ,δ=0) symmetric matter ρ ( fm 3 ) E A (ρ,β) = E A (ρ,β = 0) + S(ρ)β2 +O(β 4 ) Nuclear Matter Symmetric Matter Symmetry energy [ β = ρ ] n ρ p ρ 9

10 The Nuclear Equation of State: Infinite System neutron matter e(ρ,δ=1) e ( MeV ) 10 0 Saturation (0.16 fm 3, 16.0 MeV) S(ρ)~ -10 e(ρ,δ=0) symmetric matter ρ ( fm 3 ) E A (ρ,β) = E A (ρ,β = 0) + S(ρ)β2 +O(β 4 ) = E (J+Lx+ A (ρ,β = 1 ) 0)+β2 2 K symx 2 +O(x 3 ) [ β = ρ n ρ p ρ ; x = ρ ρ ] 0 3ρ 0 10

11 The Nuclear Equation of State: Infinite System neutron matter e(ρ,δ=1) e ( MeV ) 10 0 Saturation (0.16 fm 3, 16.0 MeV) S(ρ)~ -10 e(ρ,δ=0) symmetric matter ρ ( fm 3 ) E A (ρ,β) = E (ρ,β = 0)+β2 A ( J + L x+ 1 ) 2 K sym x 2 +O(x 3 ) S(ρ 0 ) = J d dρ S(ρ) = ρ0 L = P 0 3ρ 0 ρ 2 0 [ d2 β = ρ n ρ p ρ dρ 2S(ρ) = K sym ρ0 9ρ 2 0 ; x = ρ ρ ] 0 3ρ 0 11

12 The Nuclear Equation of State: Infinite System neutron matter e(ρ,δ=1) e ( MeV ) 10 0 Saturation (0.16 fm 3, 16.0 MeV) S(ρ)~ -10 e(ρ,δ=0) symmetric matter ρ ( fm 3 ) E A (ρ,β) = E (ρ,β = 0)+β2 A ( J + L x+ 1 ) 2 K sym x 2 +O(x 3 ) The uncertainties on S(ρ 0 ) = J d S(ρ) around saturation density dρ S(ρ) = ρ0 L = P (mainly due to L) impact on many0 nuclear physics 3ρ 0 ρ 2 d2 0 dρ 2S(ρ) and = K sym astrophysics observables. ρ0 9ρ

13 The symmetry energy and the neutron skin in 208 Pb r np r 2 n 1/2 r 2 p 1/2 Simple macroscopic model: r np 1 12 IR J L ( ) L p neut 0 r np (fm) Linear Fit, r = Mean Field SkM* Ska DD-PC1 FSUGold DD-ME1 DD-ME2 G2 Sk-T4 NL3.s25 PK1.s24 Sk-Rs RHF-PKO3 SkI2 RHF-PKA1 SV Sk-Gs NL3* NL3 PK1 NL-SV2 TM1 SkI5 G1 NL-RA1 PC-F1 NL-SH PC-PK1 NL2 NL HFB-8 MSk7 v090 SkP SkX Sk-T6 HFB-17 SGII D1N SLy5 SLy4 SkMP SkSM* SIV MSL0 MSkA BCP D1S L (MeV) Physical Review Letters 106, (2011) The faster the symmetry energy increases with density (L), the largest the size of the neutron skin in (heavy) nuclei. [Exp. from strongly interacting probes: fm (Physical Review C (2012))]. 13

14 The impact of the neutron skin on nuclear and astrophyiscs observables 14

15 The neutron skin and the parity violating asymmetry in 208 Pb 10 7 A pv 7,4 7,2 7,0 6,8 MSk7 D1S D1N SGII Sk-T6 SkP HFB-17 SkX SLy4 SLy5 SkM* BCP MSkA MSL0 DD-ME2 SIV SkMP SkSM* DD-ME1 DD-PC1 RHF-PKO3 FSUGold Zenihiro Ska Sk-Rs RHF-PKA1 Sk-Gs SV PK1.s24 SkI2 HFB-8 v090 Tarbert Hoffmann Klos Linear Fit, r = Mean Field From strong probes Sk-T4 NL3.s25 PC-PK1 G2 NL-SH NL-SV2 TM1, NL-RA1 NL3 SkI5 PC-F1 PK1 G1 NL3* 0,1 0,15 0,2 0,25 0,3 r np (fm) Physical Review Letters 106, (2011) NL2 NL1 Electrons interact by exchanging a γ (couples to p) or a Z 0 boson (couples to n) Ultra-relativistic electrons, depending on their helicity (±), will interact with the nucleus seeing a slightly different potential: Coulomb ± Weak A pv dσ+/dω dσ /dω dσ +/dω+dσ /dω Weak Coulomb Input for the calculation are the ρ p and ρ n (main uncertainty) and nucleon form factors for the e-m and the weak neutral current. In PWBA for small ( momentum transfer: A pv G Fq q2 r 2 p 1/2 ) r np 2πα 3F p(q) (Calculation at a fixed q equal to PREx) The largest the size of the neutron distribution in nuclei ( r np ), the smaller the parity violating asymmetry. [Exp. from ew probes: 0.302±0.175 fm (Physical Review C 85, (2012))]. 15

16 Isovector Giant Resonances (some considerations) In isovector giant resonances neutrons and protons oscillate out of phase Isovector resonances will depend on oscillations of the density ρ iv ρ n ρ p S(ρ) will drive such oscillations The excitation energy (E x ) within a Harmonic Oscillator approach is expected to depend on the symmetry energy: 1 d ω = 2 U m dx 2 δ k E x 2 e δβ 2 S(ρ) where β = (ρ n ρ p )/(ρ n +ρ p ) σγ abs The dipole polarizability (α Energy 2 IEWSR) measures the tendency of the nuclear charge distribution to be distorted, that is, from a macroscopic point of view α electric dipole moment external electric field 16

17 The neutron skin and the Giant Dipole Resonance in 208 Pb (E x f(0.1) S(0.1fm 3 ) ) 6.4 f(0.1)={s(0.1)(1+k}} 1/2 [MeV 1/2 ] E -1 [MeV] Physical Review C 77, (2008) The larger the neutron skin of 208 Pb, the faster the symmetry energy increases with density around saturation [ S(ρ A ) J L ρ 0 ρ A 3ρ 0 ], and the smaller the excitation energy of the Giant Dipole Resonance (GDR). 17

18 The symmetry energy and the Pygmy Dipole Resonance (Pygmy: low-energy excited state appearing in the dipole response of N Znuclei) Physical Review C 81, (2010) The larger the neutron skin in 208 Pb, the faster the symmetry energy increases with density, the larger is the energy (E) times the probability (P) of exciting the Pygmy state (EWSR = E P) WARNING: we lack of a clear understanding of the physical reason for this correlation 18

19 Dipole polarizability and the neutron skin in 208 Pb 10 2 α D J (MeV fm 3 ) r=0.97 FSU NL3 DD-ME Skyrme SV SAMi TF (fm) r np Physical Review C (2012); (2013); 92, (2015) Macroscopic model: Using the dielectric theorem: m 1 moment can be computed from the expectation value of the Hamiltonian in the constrained (D dipole operator) ground state H = H+λD Assuming the Droplet Model (heavy nucleus): [ α D α bulk D 1+ 1 ] L where 5 J α bulk D πe2 A r 2 (Migdal first derived) 54 J L αexp D αbulk D α bulk 5J D By using the Droplet Model [ one can also find: ] α D J πe2 54 A r r np r coul np r surf np 2 r 2 1/2 (I I C ) For a fixed value of the symmetry energy at saturation, the larger the neutron skin in 208 Pb, the larger the dipole polarizability. 19

20 IV-IS GQRs and the neutron skin in 208 Pb Within the Quantum Harmonic Oscillator approach E IV x = 2 hω h 2 V sym r m ( hω 0 ) 2 r 4 and EDF calculations, one can 2 deduce V sym 8(S(ρ A ) S kin (ρ 0 )) S kin 0 (ρ 0 ) ε F0 /3 (Non-Rel) E (MeV) { S(ρ A ) J L ρ 0 ρ A ε F 0 A 2/3 [ ( ) 2 ( ) ] } 2 3ρ 0 3 8ε 2 E IV x 2 E IS x +1 F 0 The larger the neutron skin in 208 Pb, the smallest the difference between the IS and IV excitation energies in GQRs. B(E2;IV) (10 3 fm 4 MeV -1 ) 6 20

21 E1 transitions in CER and the neutron skin in 208 Pb 1 -,T ,1 -,2 - IVSGDR,T ,T ,T 0 AGDR GT IAS T Z =T 0-1 Daughter nucleus L=1 L=1, S=1 S=1 strong (p,n) E1 M1 Target nucleus 1 -,T 0 1 +,T 0 T Z =T 0 0 +,T 0 E AGDR -E IAS (MeV) Exp. Data from Ref. [56] Exp. Data from Ref. [53] R n -R p (fm) Phys. Rev. C 92, (2015) AGDR ( J π = 1 with L = 1 and S = 0) is the T 0 1 component of the charge-exchange of the GDR. 5 J E AGDR E IAS 5 3 I 1+γ α H Z hc m r 2 1/2 [ ( 1 ε F 3J ) I 3 2 E AGDR E IAS ε (E E IVGDR ε) magdr 0 C m IVGDR 0 ( Rnp R surf ) ] np r 2 1/2 3 7 I C The larger the neutron skin in 208 Pb, the smaller the excitation energy of the IVGDR (as we have seen) and consistently the smaller the difference between the excitation energies of AGDR - IAS 21

22 Relevance of the neutron star crust on the star evolution and dynamics (brief motivation) The crust separates neutron star interior from the photosphere (X-ray radiation). The thermal conductivity of the crust is relevant for determining the relation between observed X-ray flux and the temperature of the core. Electrical resistivity of the crust might be important for the evolution of neutron star magnetic field. Conductivity and resistivity depend on the structure and composition of the crust Neutrino emission from the crust may significantly contribute to total neutrino losses from stellar interior (in some cooling stages). A crystal lattice (solid crust) is needed for modelling pulsar glitches, enables the excitation of toroidal modes of oscillations, can suffer elastic stresses... Mergers (binary systems that merge) may enrich the interstellar medium with heavy elements, created by a rapid neutron-capture process. In accreting neutron stars, instabilities in the fusion light elements might be responsible for the phenomenon of X-ray bursts Source: Pawel Haensel

23 The neutron skin in 208 Pb and the structure and composition of a neutron star outer crust span 7 orders of magnitude in denisty (from ionization 10 4 g/cm to the neutron drip g/cm) it is organized into a Coulomb lattice of neutron-rich nuclei (ions) embedded in a relativistic uniform electron gas T 10 6 K 0.1 kev one can treat nuclei and electrons at T = 0 K At the lowest densities, the electronic contribution is negligible so the Coulomb lattice is populated by 56 Fe nuclei. As the density increases, the electronic contribution becomes important, it is energetically advantageous to lower its electron fraction by e +(N,Z) (N+1,Z 1)+ν e and therefore Z with constant (approx) number of N As the density continues to increase, penalty energy from the symmetry energy due to the neutron excess changes the composition to a dif ferent N plateau Z A Z 0 p Fe where(a A 0 8a 0,Z 0 ) = 56 Fe 26 sym The Coulomb lattice is made of more and more neutron-rich nuclei until the critical neutron-drip density is reached (µ drip = m n). [M(N,Z)+m n < M(N+1,Z)] Composition Composition FSUGold Protons Neutrons Ni NL3 N=82 N=32 Fe Sr Sr Kr Kr N=50 N= Se Se ρ(10 11 g/cm 3 ) Sn N=82 Ge Zn Ni Cd Pd Ru Mo Zr Sr Mo Kr Zr SrKr Physical Review C 78, (2008) The larger the neutron skin of 208 Pb (L ), the more exotic the composition of the outer crust. 23

24 Some available constraints on J and L from terrestrial experiments and astrofisical observations 24

25 CONCLUSIONS 25

26 Conclusions: EoS around saturation The isovector channel of the nuclear effective interaction is not well constrained by current experimental information. Many observables available in current laboratories are sensitive to the symmetry energy. Problems: accuracy and model dependent analysis. Systematic experiments may help. Exotic nuclei more sensitive to the isovector properties (due to larger neutron excess). Problems: more difficult to measure, accuracy and model dependent analysis. Systematic experiments may help. The most promissing observables to constraint the symmetry energy are the neutron skin thickness and the dipole polarizability in medium and heavy nuclei. 26

27 Thank you for your attention! 27